6533b872fe1ef96bd12d43e1

RESEARCH PRODUCT

Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem

Pedro FreitasBarbara BrandoliniCristina TrombettiCarlo Nitsch

subject

SecondaryMathematics(all)General MathematicsEigenvalue010102 general mathematicsMathematical analysisPerturbation (astronomy)SaturationMathematics::Spectral TheoryCritical value01 natural sciencesCritical point (mathematics)010101 applied mathematicsDirichlet eigenvalueShape optimizationSettore MAT/05 - Analisi MatematicaDirichlet laplacianBall (bearing)Rayleigh–Faber–Krahn inequality0101 mathematicsNonlocalPrimaryEigenvalues and eigenvectorsMathematics

description

Abstract We determine the shape which minimizes, among domains with given measure, the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. We show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a (finite) critical value of this weight, and then remains constant. This critical point corresponds to a transition between optimal shapes, from one ball as in the Faber–Krahn inequality to two equal balls.

yearjournalcountryeditionlanguage
2011-11-01Advances in Mathematics
https://doi.org/10.1016/j.aim.2011.07.007